I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.

Consider the ODE y''+ay'+by=0 where a and b are constants such that $a^2$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $\displaystyle{\lim_{x \to \infty} Yg=0}$

So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.

So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $y=C1e^(r1*x) + C2e^(r2*x)$ where t1 and t2 are the roots to the equation $r^2+ar+b=0$ (from above equation).

The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.

$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$

So in this case I believe my Yg would be of the form $Yg = e^(r1*x) + e^(r2*x)$ which implies that my lim equation is $\displaystyle{\lim_{x \to \infty} e^(r1*x) + e^(r2*x)=0}$

I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $a^2<4b$ and that the limit of Yg is = 0.

Any guidance would be greatly appreciated.


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